Characteristics and the three gap theorem. (English) Zbl 0709.11011

van Ravenstein, Tony; Winley, Graham; Tognetti, Keith

Fibonacci Q. 28, No. 3, 204-214 (1990).

Let \([x]\) and \(\{x\}\) denote the integer and fractional parts of the positive real number \(x\). If \(d_m=[(m+1)x] - [mx]\) then \([mx]=\sum^{m-1}_{i=1}d_i+[x]\). Since each \(d_i\) is \([x]\) or \([x]+1\), we can define the characteristic of \(x\) to be the string \(x_1x_2x_3\cdots\) where \(x_i=s\) if \(d_i=[x]\) and \(x_i=\ell\) if \(d_i=[x]+1\) \((s\) for small and \(\ell\) for large). The authors exhibit a connection, for some \(x\), between the characteristic of \(x\) and the sequence of arcs or gaps formed by partitioning a circle by successive placements of points by an angle of \(x\) revolutions. They do this by making use of the three gap theorem, which asserts that points placed in this way partition a circle into gaps of 2 or 3 different lengths.

Reviewer: Ian Anderson (Glasgow)

MSC:

11K06	General theory of distribution modulo \(1\)
11A55	Continued fractions
11B83	Special sequences and polynomials

Keywords:

greatest integer; fractional parts; three gap theorem